JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Failure of geometric optics for analysis of circular optical fibers

Allan W. Snyder, D. J. Mitchell, and Colin PaskInstitute of Advanced Studies, Department of Applied Mathematics, Australian National University,

Canberra, ACT, 2600, Australia(Received 18 August 1973)

In addition to rays that lose energy by undergoing refraction, there is a large class of weakly attenuatedrays in circular optical fibers. These leaky rays are incorrectly predicted to be lossless by Fresnel's laws.Thus, Fresnel's laws fail for the analysis of long fibers. The significance and properties of leaky rays arediscussed. A very simple attenuation coefficient is given, from which the loss of all rays is computed. Thisattenuation coefficient makes it possible to extend the use of ray tracing and Snell's laws for analyzingcircular optical fibers.

Index Headings: Fiber optics; Geometric optics; Waveguides; Surface-guided waves.

It has long been widely accepted that geometric optics,i.e., ray tracing 'using Snell's laws and Fresnel's re-flection coefficients, provides a good approximation ofthe light-acceptance and transmission properties ofoptical fibers when the ratio of fiber radius to wave-length is large. For example, on p. 36 of his authori-tative account of fiber optics, Kapany' stated that thegeometric optics of fibers "is rigorously valid for calcu-lations of optical characteristics of large-diameterfibers." This assumption has led to the wide-spreadbelief'-' that a ray is transmitted, unattenuated, withina lossless fiber if it strikes the fiber boundaries at anangle equal to or greater than the critical angle neces-sary for total internal reflection. In other words, allthe rays that fall within the generalized numericalaperture' of the fiber are assumed to be trapped. Sup-port for these assumptions comes, to take an example,from the accurate prediction by geometric optics ofthe dark band across the exit end of the fiber ob-served at steep angles when the fiber is illuminatedwith a diffuse source." 4

The objectives of this paper are (i) to emphasizethat many of the so-called trapped or totally reflectedrays are in fact leaky, i.e., they lose or radiate theirenergy as they propagate within the fiber, (ii) todescribe the significance and properties of these leakyrays in physical problems, and (iii) to present numericalvalues of their attenuation. Since leaky rays accountfor a significant portion of the light-carrying capacityof the fiber, geometric optics is useful only throughouta length of fiber for which these rays retain most oftheir initial energy. We determine this length.

EXISTENCE OF LEAKY RAYSON CIRCULAR FIBER

All energy that is transmitted unattenuated in adielectric rod must he contained in bound modes,', 6

The characteristic propagation constant 3 for boundmodes traveling along the axis of a dielectric waveguidehas a minimum value 3 =27rn2/X at cut off,"- 8 where n2is the refractive index of the outside medium, as shown

in Fig. 1, and X is the wavelength in vacuum. A is theprojection of the wave vector within the fiber ontothe z axis, i.e.," 7

/27Ao= ( )coso", (1)

where O is illustrated in Fig. 1. Thus, at cut off, theangle 0, equals A,, where

n2coso 0=-. (2)

nl

The angle 0, is the complement of the critical angle

22p

I

P

PR

RD

P

FIG. 1. Illustration of angles defined at incidence on fiberboundary. P is the point of incidence. 0 is the center of thecircular cross section. ON is the angle between the normal to theboundary, given by line OP, and the incident-ray direction (RD).The angles ON, O°, and 0 are related by sino, sinos = cosON. PRis the projection of the ray direction onto the cylinder crosssection.

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ANALYSIS OF OPTICAL FIBERS

necessary for total internal reflection.' Thus, whenapplied in the appropriate asymptotic limit, bound-mode theory tells us that rays with 0,> 0c are nottrapped. This agrees with geometric optics for a planaror slab waveguide.5 6' 8 However, because rays can beskew to the circular-fiber axis, i.e., have 0kR?7r/ 2 inFig. 1, many rays with 02>00 have ON>{(ir/2)-O,}and are predicted by geometric optics to be trapped.'Therefore, in a circular fiber, all rays with 0,>0, thatare predicted by geometric optics to be trapped arein fact not trapped, i.e., there are no bound modeswith 0,> 0c.

The method for determining the existence andproperties of nontrapped rays is to solve -for certaincomplex roots of the eigenvalue equation.6' 9'" Thesecomplex roots are associated with modes that attenuateon lossless structures. Using the Debye asymptoticexpansion for Bessel functions, we can show that thefields of a mode are formed by a family of rays; eachray of the family is incident on the boundaries at thesame angle to the normal. This decomposition enablesus to study ray behavior within optical fibers. Theresults of this procedure are discussed here. By rayswe mean the trajectories of the normals to local planewaves. See Ref. 15 for mathematical details.

A. Rays in the Slab (Refs. 5, 6, and 8)

There are two main classes of rays in the slab,(i) trapped and (ii) refracted. The rays with 0,< 0 aretrapped by total internal reflection. The rays with02> 0, are attenuated by the well-known mechanism ofrefraction loss.

B. Rays in the Cylinder

There are three main classes of rays in the cylinder,(i) trapped, (ii) refracted, and (iii) leaky. The rayswith 02<00 are trapped. The rays that make an angleON< { (7r/ 2 )- O0 with the normal attenuate by re-fraction. The third class of rays have 0,>0,, withON> ( (Xr/2) - }J. These rays are trapped according togeometric optics; however, we now know that theyleak a small amount of energy. We call this third classof rays leaky rays. We refer the reader having difficultyvisualizing skew-ray propagation in an optical fiberto Ch. 2 of Ref. 1.

PROPERTIES AND SIGNIFICANCEOF LEAKY RAYS

Although most of what has been stated above is inthe literature in one form or another,',', 7"i0 the behaviorand significance of leaky rays and the consequence oftheir presence in any physical problem have remainedunappreciated or misunderstood. The remainder of thispaper is devoted to the task of filling this gap.

The most-general, descriptive property of leaky raysis that they are more skew to the fiber axis than thetrapped rays.

A. Importance of Leaky Rays

The power initially contained in the leaky raysdepends on the source of illumination. Here we supposethat all rays are launched with equal power, as is thecase for incoherent or diffuse illumination.'' 0 Ourobjective is to determine the power contribution ofleaky rays to the total power of rays predicted to betrapped by geometric optics. The power predicted bygeometric optics to be trapped within the cylinder is asum of the power of trapped rays PTR plus the initialpower of the leaky rays PLR-

The amount of power PTR transmitted by trappedrays is proportional to the square of the numericalaperture of the fiber for meridional rays [Ref. 1, Eq.(2.5)]. Thus

PTR = sin 200

where from Eq. (2)

s 1 2( 2sin 20C= J-t-

(3)

(4)

The summed power PTOT transmitted by the leaky andtrapped rays is proportional to the square of thenumerical aperture of the fiber for all skew rays,ignoring the refracted rays [Ref. 1, Eq. (2.39), withn= no]. Thus, assuming a unit power source,

PTOT= PTR+PLR

= 1--[(d-62)1+(1-25) cos'lV],7r

(5)

where we have used the notation B= 1- (n2/nl)2 . Thepercent leaky-ray power (% PLR) contribution to thetotal power predicted to be trapped by geometric opticsis

PLR% PLR= 10X

PTR+PLR

(6)

and is illustrated in Fig. 2 vs sin200 . From this figure, wecan determine the contribution of leaky rays to the totallight-transmitting property of the fiber, assuming adiffuse or incoherent illumination. When nl n2, asit is for optical-communication fibers,3 "' nearly. 50%of the power is initially launched into the leaky rays.The PTOT curve shows that, as the difference betweennl and n2 increases, the total power-transmissioncapability of the fiber increases.

We can easily construct illumination conditions forwhich only leaky rays are launched. We give threewell-known examples.

(i) The light accepted by an optical fiber, due toillumination at angles 02 of incidence greater than 00but with ON> { (7r/ 2 )-A0), is transmitted as leakyrays. " 4,12

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SNYDER, MITCHELL, AND PASK

10 0-9 08 07 06 0-5 03 l50 , I I I I I lI~ 1

IR

40

30

20

10

0o0 0 2 0 4 0-6 08

sin2 ec - 1 - (n2/n,)2

1.0

FIG. 2. Percent power % PLR carried by the leaky rays withinan optical fiber excited by an incoherent or diffuse source S. Thetotal power Pw0t is also shown. Upper scale refers to n2/nl.

(ii) The bright bands across the exit end of a fiberobserved at steep angles are due to leaky rays.'. 4

(iii) The so-called whispering-gallery modes inlarge-diameter optical fibers launched by highly skewillumination, are formed by leaky rays.'3

These examples should indicate to the reader thatthe rays we classify as leaky have an important rolein many physical problems. Until now, we have con-sidered only the amount of power accepted by theleaky rays and not the consequence of attenuation oftheir power. Nevertheless, the success of geometricoptics in describing the light-transmission phenomenain the examples cited illustrates that, in many situ-ations, the attenuation is so slight that the leakage canbe neglected. Indeed, for such examples, the limitX -> 0 of mode theory provides an inadequate descrip-tion of light transmission. This is because, in a formalelectromagnetic solution to such problems, modesaccount only for the unattenuated or guided energy,whereas radiation accounts for the leaky and refractedenergy.5 8 On the other hand, geometric optics approxi-mates certain features of the total electromagneticfield and is therefore successful when the leaky raysare only slightly attenuated.

B. Failure of Geometric Optics for Long,Multimode Fibers

That leaky rays attenuate, however slowly, indicatesthat geometric optics does not provide an adequaterepresentation of light transmission within certainvery long fibers. This phenomenon has particularrelevance to multimode liquid-core fibers proposed foruse in long-distance optical-communication systems.3 "Furthermore, the failure of geometric optics for verylong fibers explains the discrepancy between modetheory and geometric optics that appeared in the

analysis of coherent"2 and incoherently illumination of10 semi-infinite optical fibers. For certain very long fibers,

only trapped rays are significant and mode theory0*8 provides an accurate description. All rays can then be

treated as if they are meridional and ray tracing isPTOT simplified.10, 12

C. Multimode Optical Fibers of Arbitrary Length

04 Clearly there are fibers for which analysis by eithergeometric optics or mode theory is unsuitable. Forthese fibers, we need to know the attenuation of the

02 leaky rays. Knowing the attenuation of each ray, wecan preserve geometric-optic concepts for analysis of a

A fiber.

ATTENUATION OF LEAKY ANDREFRACTING RAYS

Here we present a simple analytic expression for theattenuation of all leaky rays and the refracting rayswhen the refractive index of the fiber is only slightlymore than that of its surround, i.e., when

0C-sinOe: [1-(I- 2 <<1. (7)

In most applications we are concerned with rays thatare nearly z directed, i.e.,

,-_sin0,<<1. (8)

These conditions are not restrictive, because Oz can bemany multiples of Oc and all the general ray behavior isretained. Furthermore, many of the leaky rays arecontained within Oz< 100, as illustrated by Fig. 3.Since n1-n 2 , the results are insensitive to the polar-ization of the electric vector.', 8

The power P (z) of an attenuated ray at a distance zalong the fiber axis is

P(z) =P(0)e-zlP, (9)

90 ' /286 >

6/ sin O. A% 3

RR -0-

200 1 T/8

00 00 1 2 4 6 8 10

FIG. 3. The ray type associated with 00 and 0, for the casewhen 0, and 0, are small. LR=leaky rays, TR=trapped rays,and RR=refracted rays.

A

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ANALYSIS OF OPTICAL FIBERS

where p is the fiber radius and the dimensionless attenu-ation coefficient a is found from9', 5

a 4(0\2(1\ 1

09 Ar H) } i(Q)H (Q)-

The parameters 1, Q, and V are

1= V cos0o,

F /6 2 1

-27rpn1V= 0-ti.

(10)

(11a)

(lib)

(1ic)

The angles 02 and 0, are defined by Fig. 1 and 00 isgiven by Eq. (7). The modulus of the Hankel function,IHI(Q) , is a smooth decreasing function of Q. When-ever Q or l is large, the Hankel-function product inEq. (10) can be replaced by I Hi(Q) 12.

For a given V, a/0, is determined by specifying02/00 and 0,D. In Fig. 3 we show the ray types associatedwith these two parameters. Equation (10) simplifies inthe special cases to be considered next.

APPROXIMATIONS AND NUMERICAL RESULTS

A. The Attenuation Picture in General

Figures 4 (with V= 100) and 5 (with V= 104) presentthe general behavior of the attenuation coefficient forthe leaky and refracting rays. A comparison of the

ec

40

30

20

10

2 3 4 5 6

ez/ec

FIG. 5. Same as Fig. 4 with V= 104.

figures shows the effect of approaching the geometric-optics limit, i.e., the effect of increasing V defined byEq. (11c). In particular, as V - o the attenuation ofleaky rays decreases to zero, whereas the attenuationof most of the refracted rays remains unchanged.

The attenuation coefficient associated with the curvesA, B, and C in the figures can be approximated by

0a Cc0 02

for curve A,

-=z .59( i I

(12a)

(12b)

for curve B, i.e., for rays at the critical angle, and

ao 0\2/0 - 0o Vlot/c-h2_ -V

0c, 0z+0c

for curve C.

B. Attenuation of Refracted Rays

(12c)

When a ray is incident on the fiber boundary atthan the critical angle, Fresnel's laws show that itan attenuation due to refraction loss given by"5

lesshas

(13)

1 2 3- 4 5 '6 This equation is valid unless the ray is very close to thecritical angle.

FIG. 4. The attenuation coefficient divided by A, vs 0,/Oc forV= 100. LR=leaky rays, RR = refracted rays. Curves A, B andC are associated with the approximate expressions, Eqs. (12a),(12b), and (12c), respectively. Curve A is associated with l=0,i.e., the meridional ray. Curve C is the most skew ray.

C. Transition Region Between Leakyand Refracted Rays

As shown by Figs. 6 and 7, a changes rapidly in thetransition region between leaky and refracting rays.

40

30

20

10

0

May 1974 611

0 E

a 0 2 0 2 1

-2 sin'00 - I .0C

SNYDER, MITCHELL, AND PASK

10

8OCEc

6

4

2

n'2 22 2-4 2-6 2-8

(/e

cos O

ocL

FIG. 6. Attenuation of rays near the critical angle. 0,=25.8'(cosO = O.9). The V = 104 curve obeys Fresnel's laws (FL) exceptclose to the critical angle. The critical angle is given by the verticalline. LR is leaky rays, and RR refracting rays.

The critical angle is represented by the vertical linesin the figures. This is the angle at which O. sinOp-0,Oor ON (defined by Fig. 1) is (r/ 2)- O,. The attenuationcoefficient, exactly at the critical angle, is given byEq. (12b).

The refracted rays are to the right of the criticalangle. There is a very narrow region for which therefracted rays depart from Fresnel's laws when V islarge. This is the region where Fresnel's laws, i.e.,Eq. (13), give incorrect results for refracted rays, e.g.,Fresnel's laws give a=0 for rays at the critical angle.

The leaky rays are to the left of the critical angle.Their attenuation decreases rapidly as we move awayfrom the critical angle, i.e., as either Oz or 0, decrease.Furthermore, the attenuation of leaky rays dependsdramatically on V, as we will show in further detail.We are reminded that Fresnel's laws give a = 0 forleaky rays.

D. Attenuation of Leaky Rays

Figures 8-11 provide further detail of leaky-rayattenuation. Except when a ray is very close to the

critical angle, the attenuation is extremely smallcompared with that of refracted rays. However, forlong fibers, this attenuation can be significant. Forexample, the power of a ray decreases to one-half of

e, (dog)

FIG. 7. Same as Fig. 6, but with (0/O6) = 1.35,and with varying 0>.

C

'- 1-2 1-4 1.6 1-8 20 2-2

FIG. 8. Attenuation of leaky rays. 0O0=25.8' (coso,=0.9). Theray is at the critical angle when Oz/O@= 2.294.

Vol. 64612

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ANALYSIS OF OPTICAL FIBERS

Cos 0,10 0-9 &8 0-7 0-6

iod' I I I II0-5 0A4 0-3

e,0 (deg)

FIG. 9. Attenuation of leaky rays. The heavy dashed curverepresents the critical angle.

its initial value when

z-cp

(14)

Let us calculate this a when z is equal to 103 m andthe fiber radius is p = 10-5 m. Then,

a O.7 /p\ 0.7ac Ac Z) oc(15)

where we have determined a/0C because that is how theresults are presented in Figs. 8 and 9. Typically,0-0.2, so that a ray on this fiber with a/0c of order

10(

oo

FIG. 10. Attenuation of rays when V = 500. The vertical bars on thecurves represent the ciritical angle given by 0° sin0e= O¢.

0O8 06 0-4 0-2 0

0 30 40 50 60 70 80 90

8,o (deg)

FIG. 11. Attenuation of rays vs 0d when V = 500. Verticalbars indicate the critical angle, as in Fig. 10.

10-s will have its power reduced by half. Rays on thisfiber with a/C,< 10-"1 require a length greater than106 m to decrease to half power, so that their attenu-ation is negligible. The p and 0C assumed in the fore-going are typical of proposed multi-mode liquid-coreoptical fibers"' when V- 100.

Throughout most of the leaky-ray region, the attenu-ation coefficient, as given by Eq. (10), is accuratelyapproximated by

-_2R 2 (1 -R 2 sin2C+)§tc

(16)Xexp[ -- V 1L3 R-1

where R = =O/C.

MECHANISM FOR ATTENUATIONOF LEAKY RAYS

What causes the attenuation of leaky rays? Thatthere are no leaky rays on slabs (all of the attenuatedrays are refracted rays) reveals that the curvature ofthe fiber is responsible. However, it cannot be curvaturealone or else the trapped skew rays would also beattenuated.

Marcuse' 4 has shown that radiation loss arisesbecause a portion of the energy associated with the

May 1974 613

SNYDER, MITCHELL, AND PASK

incident ray in medium 1 must also travel in medium 2.The wave in medium 2 must remain in step with thatin medium 1. Consequently, when the surface is curved,the phase velocity of the wave in medium 2 mayexceed the velocity of a plane wave in medium 2. Atthe position where this occurs, the power disassociatesitself from the incident ray, i.e., it radiates into space.This position is found from electromagnetic theory tobe at a radial distance r from the fiber axis, where forleaky rays r=p sinO, cos0o/(sin2 0,-O,2)1 and r=p forrefracted rays.

DISCUSSION

Fresnel's laws predict zero loss for all leaky rays andgive an incorrect loss for refracted rays very near thecritical angle. We have found the attenuation co-efficient for all leaky rays and those refracted rays thatare weakly attenuated. That attenuation coefficientmakes it possible to extend the use of ray tracing andSnell's laws for analyzing optical fibers. The power ofeach ray is determined from the attenuation coefficientby use of Eq. (9). Thus, a given by Eq. (10) provides ageneralized attenuation coefficient that accounts com-pletely for the losses of all rays on a lossless fiber when0, and O are small. Although the refracted rays areusually attenuated rapidly, many leaky rays persistover enormous distances. Mode theory is inadequateunless all leaky rays have been significantly attenuated.On the other hand, geometric optics, using Fresnel'slaws, can be useful only for short fibers, i.e., fibers for

which leaky-ray attenuation is negligible. In practice,a is well approximated by Eq. (16).

ACKNOWLEDGMENTS

We thank Peter McIntyre and Rowland Sammut forparticipating in numerous discussions that helped toclarify the subject of this paper. We gratefully acknowl-edge financial support by the Australian Post Officeand the stimulation of their optical group, particularlyMr. Graeme Kidd.

REFERENCES

'N. S. Kapany, Fiber Optics (Academic, New York, 1967),p. 31.

2V. Maxia, M. Murgia, and K. Testa, Appl. Opt. 12, 98(1973).

3J. P. Dakin, W. A. Gambling, H. Matsumra, D. N. Payne,and H. R. D. Sunak, Opt. Commun. 7, 1 (1973).

4R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).'R. E. Collin, Field Theory of Guided Waves (McGraw-Hill,

New York, 1960), p. 470.'N. S. Kapany and J. J. Burke, Optical Waveguides

(Academic, New York, 1972), p. 7.7A. W. Snyder, IEEE Trans Microwave Theory Tech. 17,

1130 (1969).'D. Marcuse, Light Transmission Optics (Van Nostrand

Reinhold, New York, 1972), p. 305.'A. W. Snyder and D. J. Mitchell, Electron. Lett. 9, 437

(1973).'"A. W. Snyder and C. Pask, J. Opt. Soc. Am. 63, 806 (1973)."R. D. Maurer, Proc. IEEE 61, 452 (1973).12A. W. Snyder, C. Pask, and D. J. Mitchell, J. Opt. Soc. Am.

63, 59 (1973)."F. G. Reick, Appl. Opt. 4, 1395 (1965).'4D. Marcuse, J. Opt. Soc. Am. 63, 1372 (1973).'5A. W. Snyder and D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).

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